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SingularValueDecomposition (Commons Math 3.2 API)
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org.apache.commons.math3.linear</FONT>
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Class SingularValueDecomposition</H2>
<PRE>
<A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">java.lang.Object</A>
  <IMG SRC="../../../../../resources/inherit.gif" ALT="extended by "><B>org.apache.commons.math3.linear.SingularValueDecomposition</B>
</PRE>
<HR>
<DL>
<DT><PRE>public class <B>SingularValueDecomposition</B><DT>extends <A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</A></DL>
</PRE>

<P>
Calculates the compact Singular Value Decomposition of a matrix.
 <p>
 The Singular Value Decomposition of matrix A is a set of three matrices: U,
 &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>. Let A be
 a m &times; n matrix, then U is a m &times; p orthogonal matrix, &Sigma; is a
 p &times; p diagonal matrix with positive or null elements, V is a p &times;
 n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
 p=min(m,n).
 </p>
 <p>This class is similar to the class with similar name from the
 <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
 following changes:</p>
 <ul>
   <li>the <code>norm2</code> method which has been renamed as <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getNorm()"><CODE>getNorm</CODE></A>,</li>
   <li>the <code>cond</code> method which has been renamed as <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()"><CODE>getConditionNumber</CODE></A>,</li>
   <li>the <code>rank</code> method which has been renamed as <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getRank()"><CODE>getRank</CODE></A>,</li>
   <li>a <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()"><CODE>getUT</CODE></A> method has been added,</li>
   <li>a <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()"><CODE>getVT</CODE></A> method has been added,</li>
   <li>a <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSolver()"><CODE>getSolver</CODE></A> method has been added,</li>
   <li>a <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getCovariance(double)"><CODE>getCovariance</CODE></A> method has been added.</li>
 </ul>
<P>

<P>
<DL>
<DT><B>Since:</B></DT>
  <DD>2.0 (changed to concrete class in 3.0)</DD>
<DT><B>Version:</B></DT>
  <DD>$Id: SingularValueDecomposition.java 1456931 2013-03-15 12:34:35Z luc $</DD>
<DT><B>See Also:</B><DD><a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>, 
<a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a></DL>
<HR>

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<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#SingularValueDecomposition(org.apache.commons.math3.linear.RealMatrix)">SingularValueDecomposition</A></B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculates the compact Singular Value Decomposition of the given matrix.</TD>
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<CODE>&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()">getConditionNumber</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Return the condition number of the matrix.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getCovariance(double)">getCovariance</A></B>(double&nbsp;minSingularValue)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the n &times; n covariance matrix.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getInverseConditionNumber()">getInverseConditionNumber</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Computes the inverse of the condition number.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getNorm()">getNorm</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the L<sub>2</sub> norm of the matrix.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;int</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getRank()">getRank</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Return the effective numerical matrix rank.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getS()">getS</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the diagonal matrix &Sigma; of the decomposition.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;double[]</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSingularValues()">getSingularValues</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the diagonal elements of the matrix &Sigma; of the decomposition.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/DecompositionSolver.html" title="interface in org.apache.commons.math3.linear">DecompositionSolver</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSolver()">getSolver</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Get a solver for finding the A &times; X = B solution in least square sense.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getU()">getU</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the matrix U of the decomposition.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()">getUT</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the transpose of the matrix U of the decomposition.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getV()">getV</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the matrix V of the decomposition.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()">getVT</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the transpose of the matrix V of the decomposition.</TD>
</TR>
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<A NAME="SingularValueDecomposition(org.apache.commons.math3.linear.RealMatrix)"><!-- --></A><H3>
SingularValueDecomposition</H3>
<PRE>
public <B>SingularValueDecomposition</B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix)</PRE>
<DL>
<DD>Calculates the compact Singular Value Decomposition of the given matrix.
<P>
<DL>
<DT><B>Parameters:</B><DD><CODE>matrix</CODE> - Matrix to decompose.</DL>
</DL>

<!-- ============ METHOD DETAIL ========== -->

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<A NAME="getU()"><!-- --></A><H3>
getU</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getU</B>()</PRE>
<DL>
<DD>Returns the matrix U of the decomposition.
 <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the U matrix<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()"><CODE>getUT()</CODE></A></DL>
</DD>
</DL>
<HR>

<A NAME="getUT()"><!-- --></A><H3>
getUT</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getUT</B>()</PRE>
<DL>
<DD>Returns the transpose of the matrix U of the decomposition.
 <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the U matrix (or null if decomposed matrix is singular)<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getU()"><CODE>getU()</CODE></A></DL>
</DD>
</DL>
<HR>

<A NAME="getS()"><!-- --></A><H3>
getS</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getS</B>()</PRE>
<DL>
<DD>Returns the diagonal matrix &Sigma; of the decomposition.
 <p>&Sigma; is a diagonal matrix. The singular values are provided in
 non-increasing order, for compatibility with Jama.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the &Sigma; matrix</DL>
</DD>
</DL>
<HR>

<A NAME="getSingularValues()"><!-- --></A><H3>
getSingularValues</H3>
<PRE>
public double[] <B>getSingularValues</B>()</PRE>
<DL>
<DD>Returns the diagonal elements of the matrix &Sigma; of the decomposition.
 <p>The singular values are provided in non-increasing order, for
 compatibility with Jama.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the diagonal elements of the &Sigma; matrix</DL>
</DD>
</DL>
<HR>

<A NAME="getV()"><!-- --></A><H3>
getV</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getV</B>()</PRE>
<DL>
<DD>Returns the matrix V of the decomposition.
 <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the V matrix (or null if decomposed matrix is singular)<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()"><CODE>getVT()</CODE></A></DL>
</DD>
</DL>
<HR>

<A NAME="getVT()"><!-- --></A><H3>
getVT</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getVT</B>()</PRE>
<DL>
<DD>Returns the transpose of the matrix V of the decomposition.
 <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the V matrix (or null if decomposed matrix is singular)<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getV()"><CODE>getV()</CODE></A></DL>
</DD>
</DL>
<HR>

<A NAME="getCovariance(double)"><!-- --></A><H3>
getCovariance</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getCovariance</B>(double&nbsp;minSingularValue)</PRE>
<DL>
<DD>Returns the n &times; n covariance matrix.
 <p>The covariance matrix is V &times; J &times; V<sup>T</sup>
 where J is the diagonal matrix of the inverse of the squares of
 the singular values.</p>
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>minSingularValue</CODE> - value below which singular values are ignored
 (a 0 or negative value implies all singular value will be used)
<DT><B>Returns:</B><DD>covariance matrix
<DT><B>Throws:</B>
<DD><CODE><A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</A></CODE> - if minSingularValue is larger than
 the largest singular value, meaning all singular values are ignored</DL>
</DD>
</DL>
<HR>

<A NAME="getNorm()"><!-- --></A><H3>
getNorm</H3>
<PRE>
public double <B>getNorm</B>()</PRE>
<DL>
<DD>Returns the L<sub>2</sub> norm of the matrix.
 <p>The L<sub>2</sub> norm is max(|A &times; u|<sub>2</sub> /
 |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm
 (i.e. the traditional euclidian norm).</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>norm</DL>
</DD>
</DL>
<HR>

<A NAME="getConditionNumber()"><!-- --></A><H3>
getConditionNumber</H3>
<PRE>
public double <B>getConditionNumber</B>()</PRE>
<DL>
<DD>Return the condition number of the matrix.
<P>
<DD><DL>

<DT><B>Returns:</B><DD>condition number of the matrix</DL>
</DD>
</DL>
<HR>

<A NAME="getInverseConditionNumber()"><!-- --></A><H3>
getInverseConditionNumber</H3>
<PRE>
public double <B>getInverseConditionNumber</B>()</PRE>
<DL>
<DD>Computes the inverse of the condition number.
 In cases of rank deficiency, the <A HREF="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()"><CODE>condition
 number</CODE></A> will become undefined.
<P>
<DD><DL>

<DT><B>Returns:</B><DD>the inverse of the condition number.</DL>
</DD>
</DL>
<HR>

<A NAME="getRank()"><!-- --></A><H3>
getRank</H3>
<PRE>
public int <B>getRank</B>()</PRE>
<DL>
<DD>Return the effective numerical matrix rank.
 <p>The effective numerical rank is the number of non-negligible
 singular values. The threshold used to identify non-negligible
 terms is max(m,n) &times; ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>)
 is the least significant bit of the largest singular value.</p>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>effective numerical matrix rank</DL>
</DD>
</DL>
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<A NAME="getSolver()"><!-- --></A><H3>
getSolver</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/DecompositionSolver.html" title="interface in org.apache.commons.math3.linear">DecompositionSolver</A> <B>getSolver</B>()</PRE>
<DL>
<DD>Get a solver for finding the A &times; X = B solution in least square sense.
<P>
<DD><DL>

<DT><B>Returns:</B><DD>a solver</DL>
</DD>
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